Dalgety's scheme encompasses fourteen main classes, with several sub-classes within each main class,
while Slocum makes due with 10 main classes and few subclasses.
There is considerable overlap and good correspondence between the two systems.
The
Wikipedia entry for mechanical puzzles
also pretty much agrees with the top level classes.

The classification scheme I have employed on Rob's Puzzle Page is closer to Dalgety's since I prefer finer granularity.
I have seen various other schemes around the web, for instance, based on the material from which the puzzle is made -
e.g. plastic, wood, glass - but I don't
think they're very useful for finding a puzzle or identifying isomorphic or related puzzles.
I do think each classification scheme reveals something about the preferences, biases, experience, expertise,
and focus of the classifier.

The table below compares and attempts to align the different schemes.
I have also found it instructive to include the categories described by van Delft and Botermans in their 1978 classic
Creative Puzzles of the World, and the ten sections in Hoffmann's 1893 book, since
they help show how the universe of puzzles has evolved over time.

Where I have found expert advice in further classifying particular categories of puzzle, I have
highlighted the experts and their categories like this.

Note that a different and perhaps more productive approach to classifying puzzles would be tag based.
For example, a puzzle might have attributes of both assembly as well as pattern.
Its assignment to one traditional category or the other would be based on a weighting of all of its attributes as determined by an expert authority or by mass concensus.

Supposedly the jigsaw was invented in 1760 by John Spilsbury of London,
but dissected maps were being used as teaching aids in the mid 1700s.
Jigsaws for amusement appeared as early as 1785 and were available in America by the early 1800s.
Hoffmann's section III includes the Chinese Zig Zag, a 3D jigsaw, but no other jigsaw puzzles.
Most jigsaw puzzles made in the 19th century had juvenile pictures, fewer than 75 pieces, and were too simple to hold an adult's attention.
The first jigsaw craze in the U.S. did not occur until 1908.
For these reasons it is not surprising that Hoffmann omits the category.
(Some of the above factoids I learned from Anne Williams' excellent book.)

The Englishman John Walker invented and sold the first friction matches in 1827.
In 1830, Samuel Jones begain making and selling them using the name "lucifers."
By 1880 match manufacturing was a huge business.

Hoffmann indicates that even in his day there were many matchstick puzzles. He includes only "a brief selection" of 19.

As for other types of pattern puzzle,
he covers the 8 Queens puzzle in his chapter VI, #24.
Chapter X, #18 Treasure at Medinet is an 8-Queens variant like Jeu des Manifestants.
He includes several magic squares in chapter IV.
The Thurston edge matching puzzle used by Calumet, and the MacMahon/Journet Mayblox were both first patented in 1892.
Hoffmann might not yet have seen them -
the only edge-matching type puzzles he mentions are
"The Royal Aquarium Thirteen Puzzle" (equivalent to the French Le Nombre Treize)
#72 in chapter IV, and
"The Endless Chain" (equivalent to the French La Chaine sans Fin) #18 in chapter III.
The Instant Insanity family wasn't invented until 1900.

Chapter II is where Hoffmann really shines for a mechanical puzzle collector.
He lumps some tanglements in as well, and it makes a kind of sense since one has to take them apart,
but today we would classify them separately.

Hoffmann didn't spike out Interlocking puzzles -
he included a 6-piece burr in his "Combination" chapter III, #36.
Slocum has traced the 6-piece burr back to Germany in 1698.
Figural/representational Kumiki puzzles were invented in Japan in the 1890s by Tsunetaro Yamanaka, but
the world had to await the invention of many of the best geometric interlocking puzzle designs by Stewart Coffin in the 1970s.
The keychain puzzle originated at the 1939 New York World's Fair.

According to Slocum and Botermans (1994), this class contains the oldest surviving mechanical puzzles -
a puzzle jug made in the south of France circa 1400 is in the Exeter Museum.
However, Dalgety believes the oldest puzzle may be a
dexterity puzzle found in the Indus Valley
dating to circa 2500 B.C.
Another contender for the title of Oldest Mechanical Puzzle is
Archimedes' Stomachion, dating back about 2200 years.
(Read about the Stomachion palimpsest.)

Regardless of its antiquity, this is an interesting but potentially messy (and vaguely cruel) category.
While I do own one trick cup, I have not focused here.

The classification scheme used by the U.S. Patent Office
devotes a class and several subclasses to puzzles.
To review the classification scheme from the top down, begin at the page
"
US Classes by Number with Title."
For mechanical puzzles, you'll probably want
Class 273.
Go to the Advanced Search Page
to enter a query, select the year range to include in your search, and execute your search.